| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5·5-s − 6·6-s − 13.9·7-s − 8·8-s + 9·9-s − 10·10-s + 6.90·11-s + 12·12-s − 65.5·13-s + 27.9·14-s + 15·15-s + 16·16-s − 18.7·17-s − 18·18-s − 50.2·19-s + 20·20-s − 41.9·21-s − 13.8·22-s + 138.·23-s − 24·24-s + 25·25-s + 131.·26-s + 27·27-s − 55.9·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.189·11-s + 0.288·12-s − 1.39·13-s + 0.533·14-s + 0.258·15-s + 0.250·16-s − 0.267·17-s − 0.235·18-s − 0.606·19-s + 0.223·20-s − 0.436·21-s − 0.133·22-s + 1.25·23-s − 0.204·24-s + 0.200·25-s + 0.989·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.616670315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616670315\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 31 | \( 1 + 31T \) |
| good | 7 | \( 1 + 13.9T + 343T^{2} \) |
| 11 | \( 1 - 6.90T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 36.7T + 2.43e4T^{2} \) |
| 37 | \( 1 - 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 41.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 162.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 558.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 223.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 553.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 535.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 976.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 695.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 230.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 558.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 111.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.526323670232697852552782856083, −9.082998900366441525645211254573, −8.102822296827348016205063986019, −7.14904156111486223764741358851, −6.58515898331001620446724600322, −5.42608118225146330648530872568, −4.22268461330202292135041485251, −2.89349050658027393747650299492, −2.22462440950124216320454616197, −0.72649995555610872365497946158,
0.72649995555610872365497946158, 2.22462440950124216320454616197, 2.89349050658027393747650299492, 4.22268461330202292135041485251, 5.42608118225146330648530872568, 6.58515898331001620446724600322, 7.14904156111486223764741358851, 8.102822296827348016205063986019, 9.082998900366441525645211254573, 9.526323670232697852552782856083
